In general, laboratory spectrometers receive signals from one radiation source and one sample, and display the results as a single spectrum of the sample.
In many applications for sensing and process control it is desirable to use a single costly spectrometer to analyze the signals from many sources and/or samples. The simplest way to do this is to present each signal sequentially to the spectrometer, and analyze the results separately. Alternatively, a multiplexing technique in which several signals are presented and analyzed at once may be used. Multiplexing has two advantages, the total signal arriving at the detector is greater than if the signals are observed sequentially and, for systems in which the noise is independent of the signal, the overall signal-to-noise ratio is increased relative to that which can be obtained by sequential sampling. The disadvantage is that it its necessary to determine the contribution of each source and sample to the total observed (composite) signal. This may be done by encoding the information carried by each signal in some appropriate way before recording the composite signal, and then using a suitable decoding process to recover the individual signals.
A variety of encoding and decoding schemes, which differ in their degree of difficulty of implementation, are possible, Typically the observations Y.sub.i which measure the total signal presented to the decoder (data-handling system) can be related to the unknown individual signals X.sub.j by a system of linear simultaneous equations which can be written in matrix form: ##EQU1##
The unknowns X.sub.j can be recovered by solving these equations.
The equation (1) presents a unifying framework for discussing the design of various kinds of spectrometers. The simplest case is that of a dispersive instrument, such as a conventional grating infra-red spectrometer or continuous wave NMR. The values X.sub.j are radiation intensities at various wavelengths, distinguished by a wavelength sorting device such as a diffraction grating. These intensities are recorded by the instrument and output in the form of a spectrum Y.sub.i, i=1, 2, . . . , N. Thus, in this trivial case, the coefficients A.sub.ij comprise a unit matrix.
More complicated, and closely related, examples are provided by Fourier transform infra-red (FT-IR) and pulsed Fourier transform nuclear magnetic resonance (FT-NMR) spectrometers. In the former case, wavelength information (i.e. radiation intensities X.sub.j at various wavelengths) is encoded by a Michelson interferometer into a series of intensities Y.sub.i sorted as a function of the position of a movable mirror in one arm of the interferometer. In the NMR case the encoding is achieved, not by some physical device, but by the precession and progressive dephasing of the nuclear spins within the sample. Intensity values X.sub.j as a function of frequency are encoded and recorded as a series of induced voltages Y.sub.i measured at successive time intervals after the excitation of the sample.
In both of these cases, the matrix A is a unitary matrix of Fourier coefficients. Equation (1) can be solved by carrying out an inverse Fourier transform of the data Y.sub.i.
An infinite number of other possible choices for the coefficients A.sub.ij can be envisaged. However, there are four important limitations on the possible choices:
1. It must be possible to solve equation (1). The basic requirement is that the determinant of A should be non-zero. In order to obtain accurate numerical solutions, it is also desirable that A should not be ill-conditioned. PA0 2. It must be possible to realize the coefficients A.sub.ij physically in some way. The encoding process may be implemented explicitly in hardware (e.g. the Michelson interferometer), or it may be implicit in the physics of the system (e.g. nuclear precession). The hardware implementation is typically easier if the array is cyclic (i.e. each row is obtained by shifting the previous row one element to the left or right). PA0 3. It is desirable that the experimental errors or signal-to-noise ratios of the decoded X.sub.j values should be at least as good as, or better than, the errors obtained when measuring the X.sub.j values directly by some other technique. Furthermore, the Improvement in signal-to-noise ratio should preferably be the same for all the X.sub.j . This imposes a severe restriction on A. PA0 4. It is desirable (though not essential) that the matrix A can be constructed and implemented for any order N. Changing N should not require radical redesign of the instrument.
We now consider points 2 and 3 above in more detail. Consider first a simple series of N measurements X.sub.j of some quantity, such as the individual weights of a group of objects. Suppose that the errors in these measurements have standard deviation .sigma., and that the weights of arbitrary collections of these objects can also be measured, also with error .sigma.. It possible to determine the weights of the individual objects more accurately than .sigma. by weighing collections of the objects groups, rather than individually. The choice of objects in each group is made according to a suitable weighing scheme. Given N objects, N groups of these objects must be weighed according to a weighing scheme which may be represented by an N.times.N matrix of zeroes and ones. The presence of a one in the (i,j)th element indicates the inclusion of object j in the ith group, while a zero indicates its absence. In equation (1), the matrix A is the weighing scheme and the quantities Y.sub.i are the weights of the groups. For example, a suitable weighing scheme for 3 objects might look as follows: ##EQU2##
A matrix of zeroes and ones is appropriate to a type of measurement in which the individual measurements can be co-added into the group measurements, or omitted. In the weighing example, this is the situation when a spring balance is used: each object can be put on the pan, or not, as required.
If a beam balance is used, we have three options for each object: use the left-hand pan, the right-hand pan, or neither. The weighing scheme for this experiment requires a matrix of zeroes, ones and minus ones. This clearly illustrates that the nature of the experiment determines the possible values of the coefficients A.sub.ij which can be realized.
Considerable theoretical work has been carried out to find suitable matrices of the above types, which satisfy the above limitations. The conditions are very restrictive, and suitable matrices known only for certain values of N. Confining our attention to matrices of zeroes and ones, the best solutions are the so-called Hadamard simplex matrices, or S-matrices. The process of solving equation (1) is then an inverse Hadamard transform. Exactly one S-matrix (plus its cyclic permutations) exists for each N of the form N=4n-1, n=1, 2, 3, . . . (Note that requirement 4 above is not satisfied. How we proceed when N is not of this form will be dealt with below). Constructions for S-matrices of various orders satisfy requirement 3; the signal-to-noise ratio of each unknown X.sub.j (computed by inverse Hadamard transform) is improved by an enhancement factor ##EQU3## relative to the signal-to-noise ratio of X.sub.j measured directly.
These principles have been used in the construction of Hadamard transform spectrometers (HTS). For example, Decker and Harwit (J. A. Decker, Jr., and M. Harwit, Appl. Opt., 8. (1969), p. 2552) constructed a multiplexing dispersive instrument using a multislit encoding mask at the exit focal plane of a conventional dispersive monochromator. This mask allowed more than one spectral resolution element (i.e. radiation at different wavelengths) to impinge upon the detector at once; hence, the encoding mask multiplexed the dispersed radiation. The pattern of slits in the mask modelled the pattern of zeroes and ones in one row of an S-matrix. The mask was physically moved to allow a different combination of spectral resolution elements to impinge on the detector before each detector reading was performed.